Orientation: One Idea in Five Disguises

Ask a condensed-matter physicist, a field theorist, and a representation theorist what a partition function is, and you will get three answers that sound unrelated. The first will write a sum of Boltzmann weights; the second a path integral; the third a generating function counting dimensions of graded pieces of a module. These notes are an argument that they are describing the same object from different angles, and that learning to translate between the descriptions is most of what it means to move comfortably between the fields.

The one idea

The partition function is a weighted trace, which is a graded character. Its deep payoff is modular invariance, which binds the physical torus to the mathematical characters.

We will meet the same object wearing five costumes, one per chapter:

$\displaystyle\underbrace{Z=\sum_{i}\mathrm{e}^{-\beta E_{i}}}_{\text{L1: a sum% }}\;\longrightarrow\;\underbrace{Z=\operatorname{Tr}\,\mathrm{e}^{-\beta H}}_{% \text{L2: a trace}}\;\longrightarrow\;\underbrace{Z=\int\mathcal{D}\phi\;% \mathrm{e}^{-S_{E}}}_{\text{L3: a path integral}}\;\longrightarrow\;% \underbrace{Z(\tau)=\operatorname{Tr}\,q^{L_{0}-\frac{c}{24}}\,\bar{q}^{\bar{L% }_{0}-\frac{c}{24}}}_{\text{L4: a torus amplitude}}\;\longrightarrow\;% \underbrace{\operatorname{ch}V=\sum_{n}(\dim V_{n})\,q^{n}}_{\text{L5: a % character}}$

Three threads to watch

Thread (Thread 1 - the Boltzmann factor, abstracted).

The weight $\mathrm{e}^{-\beta E}$ is the most-used and least-examined object in the subject. Its abstraction - a homomorphism turning additive conserved charges into multiplicative weights, with one conjugate potential per charge - is developed in Lectures 1–2 and consolidated in the Interlude. Watch the exponent grow new terms (a chemical potential, a sign grading) and watch those conjugate potentials get renamed downstream: $\beta\to\operatorname{Im}\tau$ , fugacity $\to$ elliptic variable $y$ , $\to$ the Cartan variables of a character.

Thread (Thread 2 - the boson and the fermion).

Introduced as single modes in Lecture 1, they are promoted at each stage until, on the torus, the boson yields the Dedekind eta and the fermion’s four spin structures yield the Jacobi theta functions. That is precisely the data needed to reach superconformal field theory - NS/R sectors, the Witten index, the elliptic genus - at the close.

Thread (Thread 3 - the Casimir constant $-c/24$ ).

A single quantity surfaces three times: as the regularized vacuum energy $\zeta(-1)=-\tfrac{1}{12}$ in Lecture 3, as the $q^{1/24}$ in $\eta(\tau)$ in Lecture 4, and as the universal $L_{0}-\tfrac{c}{24}$ grading shift in Lecture 5.

Conventions, fixed once

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We work in Euclidean signature for partition functions; Lorentzian language appears only as motivation.
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The symbol $q$ changes meaning, and we will say so each time. In thermal physics a single mode contributes with $q=\mathrm{e}^{-\beta\hbar\omega}$ ; on the torus $q=\mathrm{e}^{2\pi\mathrm{i}\tau}$ . The two agree only after $\beta$ is identified with an imaginary period.
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$\beta=1/(k_{B}T)$ ; we set $k_{B}=\hbar=1$ after Lecture 2.
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$\tau=\tau_{1}+\mathrm{i}\tau_{2}\in\mathbb{H}$ , the upper half-plane, so $\operatorname{Im}\tau=\tau_{2}>0$ .
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In the 2d chapters we take $c=\bar{c}$ : one central charge for both chiralities.
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Vocabulary clashes are flagged inline: “weight” (CFT conformal weight $h$ vs. Lie-theory weight $\lambda$ ), “character” (group trace vs. graded dimension), “level” (affine $k$ vs. an energy level).

Each chapter opens with a sidebar-“For the reader from camp X”-mapping the vocabulary you already own onto the one we are about to use. Heavy machinery and proofs are quarantined in “For the curious” boxes; nothing in the main narrative depends on them.